Realizing degree sequences as -connected graphs

An integer-valued sequence π = ( d 1 , … , d n ) is graphic if there is a simple graph G with degree sequence of π . We say the π has a realization G . Let Z 3 be a cyclic group of order three. A graph G is Z 3 -connected if for every mapping b : V ( G ) → Z 3 such that ∑ v ∈ V ( G ) b ( v ) = 0 , there is an orientation of G and a mapping f : E ( G ) → Z 3 − { 0 } such that for each vertex v ∈ V ( G ) , the sum of the values of f on all the edges leaving from v minus the sum of the values of f on the all edges coming to v is equal to b ( v ) . If an integer-valued sequence π has a realization G which is Z 3 -connected, then π has a Z 3 -connected realization G . Let π = ( d 1 , … , d n ) be a nonincreasing graphic sequence with d n ≥ 3 . We prove in this paper that if d 1 ≥ n − 3 , then π has a Z 3 -connected realization unless the sequence is ( n − 3 , 3 n − 1 ) or is ( k , 3 k ) or ( k 2 , 3 k − 1 ) where k = n − 1 and n is even; if d n − 5 ≥ 4 , then π has a Z 3 -connected realization unless the sequence is ( 5 2 , 3 4 ) or ( 5 , 3 5 ) .